We employ the FeynArts, FormCalc and LoopTools packages
(versions 3.11, 9.9 and 2.16, respectively) [1]}, [2]} to
aid with the calculation and to evaluate the results numerically. Ignoring
terms of order \( m_{\mu }^2 / M_W^2\)  and \( m_{\mu }^2 / M_{Z^\prime }^2 \) , the
Lorentz structure of the \( U(1)_{B_3-L_2}\)  amplitude is identical to that of the
4-Fermi theory and we can match the the two theories at the amplitude
level. In doing so, we find that \( \delta (\Delta r) \)  is dominated by the
oblique corrections to the vector boson propagators, overwhelming the \(Z^\prime \) -induced effects by several orders of magnitude. The reason is that
all of the \( Z^\prime \) -effects are of order \( g_{Z^\prime }^2 m_{\mu }^2 / M_{Z^\prime }^2 \)  and thus become negligible for a TeV scale \( Z^\prime \) . Thus, the resulting constraint on \( \sin \phi \)  is essentially independent of \( M_{Z^\prime } \)  and \( g_{Z^\prime } \) .
